DYN AMICS OF A RIGID BODY. 173 



1 60. Theorem. When an impulsive wrench acting 

 on a free rigid body produces an instantaneous rotation, 

 the axis of the rotation must be perpendicular to the im- 

 pulsive screw. 



Let ni, . . . rj 6 be the axis of the rotation, then 



or 



whence the screw of which the co-ordinates are + 

 - 0rj>, + bv} Zy &c., is perpendicular to r/, and the theorem is 

 proved. 



From this theorem, and the last, we infer that, when 

 an impulsive force acting on a rigid body produces 

 an instantaneous rotation, the direction of the force, and 

 the axis of the rotation, are parallel to the principal 

 axes of a section of the momental ellipsoid. 



161. Principal Axis. If rj be a principal axis of a 

 rigid body, it is required to prove that 



reference being made to the absolute principal screws of 

 inertia. 



For in this case a force along a line 6 intersecting /, 

 compounded with a couple in a plane perpendicular to r\, 

 must constitute an impulsive wrench to which rj corres- 

 ponds as an instantaneous screw, whence we deduce 

 ( 93)> ^ an d k being arbitrary constants. 



. 



l -7- -T !!, 



P\ am 



&c., 



n h dR ^ 

 06 = -r -T + kpwt. 

 pt dm 



Expressing the condition that pe = o, we have 



